NAME _________________________
GEOMETRY NOTES
UNIT 6
COORDINATE GEOMETRY
DATE
Jan
2,2013
PAGE
2
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3,4
1/6
4,5
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6,7
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7
1/9
8,9
TOPIC
Solve equations for y=mx+b
Write equations given slope and y –
intercept.
Graph linear equations
Find slope given two points
Write equation of a line given two points
and y-intercept
Equations of lines parallel and
perpendicular to each other
Write the equations of lines parallel and
perpendicular through a given point.
More Equations
QUIZ
Find Midpoint and Distance
1/10
1/13
10
xx
Find Perpendicular Bisector
Practice perpendicular bisector
1/14
11
1/15
1/16
12/13
14
QUIZ
Proving things about lines
Proving Types of triangles/Trapezoid
Proving parallelograms
1/17
15
1/21
16
1/22
1/23
1/24
2/3
2/4
2/5
17
18
QUIZ
Proving Rectangles
Proving Rhombus
Proving Squares
REVIEW FOR MIDTERMS
REVIEW FOR MIDTERMS
Prove a figure is….
Review
TEST
1
HOMEWORK
Pg 169# 1, 4, 5, 6, 33, 34.
*Graph Paper*
Parallel Lines in the Coordinate
Plane Worksheet #1, 3, 4, 6, 7, 8, 9,
13, 15, 17
Pg 177# 1, 2, 6, 8, 10, 16, 18, 26
Pg 178# 12, 14, 20, 22
No HOMEWORK!
Distance/Midpoint Worksheet.
Circled Questions
NO HOMEWORK!
Finish In Class Assignment if not
done.
Pg 351 #12-16
Pg 353 #42-44
Worksheet HW tri/traps
Worksheet Prove JOHN is a
parallelogram
Worksheet Prove WARD is a
rectangle
Worksheet Prove DROW is a
rhombus
Worksheet Prove ANDY is a square
TBD
STUDY
No Homework
TICKET-IN
No Homework
Solve for y/Write equation of a line/Graph
The general equation for a line is ____=_________________________ where
m=
b=
Write the equation of a line with the given slope and y-intercept:
1.) Slope: ½
2.) Slope: -2/3
y-intercept: 4
y-intercept: -1
3.) Slope: 0
y-intercept: 3
Solve for y and state the slope and y-intercept:
4.)2y=7x-10
5.) 5x+3y =3
6.) x-6y=12
7.) Graph the following linear equation: 3y+6=x
2
SLOPE
Slope: _____________________________________________________________________________
Slope Formula:
#𝒐𝒇 𝒔𝒑𝒂𝒄𝒆𝒔 𝒖𝒑 𝒐𝒓 𝒅𝒐𝒘𝒏
OR Find it by counting # 𝒐𝒇 𝒔𝒑𝒂𝒄𝒆𝒔 𝒕𝒐 𝒕𝒉𝒆 𝒓𝒊𝒈𝒉𝒕
* You should always write slope as a ___________________ and use _________ to represent slope.
Find the slope of the line between the two points two ways, using the formula and by graphing.
Formula
Graph
The slope is:
1.) (4,8) (6,-2)
2.) (6,-2) (6,7).
3.)Write the equation of the line that pass
through (4,-5) (6,-8)
4.) Write the equation of a line that passes
through (4,-3) (-3,-3).
3
5.) a.) Using the slope formula, find the slopes of the lines passing through the following points:
(-6,1) and (3,7)
(0,-9) and (9,-3)
b.) What do you notice about their slopes?_________________________________________________
c.) Now graph the line segments:
Hmmm...looks like the lines are ____________
So ______________ ___________ always have
the _______________ slope!
6.) a.)Using the slope formula, find the slopes of the lines passing through the following points:
(-3,1) and (6,4)
(-2,8) and (1,-1)
b.) What do you notice about the slopes?__________________________________________________
c.) Now graph the line segments:
Hmmm...looks like the lines are ____________
So ______________ ___________ always have
the ____________ ________________ slope!
NOTE: There is one exception to this rule; a horizontal line is perpendicular to a vertical line but their slopes are
not negative reciprocals of each other.
Their slopes are ___________________ and __________________
4
Writing equations of lines parallel and perpendicular to given lines.
Remember:
Lines that are parallel have______________________________________________________
Lines that are perpendicular have_________________________________________________
Identify the slope of each line given then write an equation of any line that would be parallel to that line:
1.) y=2x+7
2.) 4y=x-3
3.) 2y+2=3x
Identify the slope of each line given then write an equation of any line that would be perpendicular to that
line:
4.) y=-2x+4
5.) 3y=5x-1
5
6.) 4y-1=-x
Writing equations of Parallel and Perpendicular lines through a given point
RECALL:
Slopes of parallel lines are __________________.
Slopes of perpendicular lines are ____________________ _____________________.
The general form of the equation of a line:
Given the following point and slope, write the equation of a line in standard form:
1.) m=3
2.) m=1/2
3.) m= -1
(-1,4)
(6,1)
(-3,5)
Writing equations of lines parallel:
4.) Write an equation of a line that is parallel to y= -4x+3 that contains the point (1,-2). (Try using the graph to
help).
5.) Write the equation of a line parallel to x=8 and goes through (4,-2)
6
6.) Write an equation of a line that is parallel to 6x-3y =9 that contains the point (-5,-8).
Writing equations of lines perpendicular:
7.) Write an equation for the line through (-3,7) and perpendicular to y = -3x-5.
8.) Write the equation of a line that is perpendicular to x=8 and goes through (4,-2)
9.) Write and equation for the line that contains (10,0) and is perpendicular to 5x+2y=1.
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MIDPOINT
Midpoint of a line segment ____________________________________________________________
Midpoint Formula: The midpoint M of AB with endpoints A(x1,y1) and B(x2,y2) are the following:
Graph the points of each segment and find the midpoints:
1.) Q(3,5) and S(7,-9)
2.) A(6,0) and B(4,-4)
Hmm… looks like the
lines _______________
each other!
RULE: If two lines have the same midpoint then the lines _______________ ________ ___________.
3.) The midpoint of AB is M(3,4). One of the endpoints is A(-3,-2). Find the coordinates of the other endpoint
B.
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FINDING THE DISTANCE BETWEEN TWO POINTS
Find the distance between the two points below. Use the graph to help you.
1.) ( -2,5) and (3,5)
2.) (3,5) and (3,1)
3.) (-2,5) and (3,1)
Find the distance between the two points:
4.) (-6,2) and (4,8)
5.) (0,7) and (-10, 3)
RULE: If two line segments have the same distance then they are _____________________________.
Note: When finding the distance between two points
you can either use the Pythagorean Theorem or the distance formula.
9
PERPENDICULAR BISECTOR
Remember:
Equation of a line:___________ where m=__________________ and b=_____________________
Perpendicular lines have:____________________ ______________________ _____________________
To bisect a line you find the _______________________________
If the endpoints of a line segment are given as (-3,5) and (7,7), find the perpendicular bisector of the line
segment:
Follow these steps:
1.) Find the slope of the line segment:
2.) Write the slope of a line that would be perpendicular to that line:
3.) What is the midpoint of the segment?
4.) What is the y-intercept of a line with the slope from #2 and
that goes through the point from #3?
5.) What is the equation of a line with the slope from #2 and the y-intercept from #4?
Now you try:
6.) Find the perpendicular bisector of the segment joining the
points (-10,-4) and (-2,0)
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PROVING THINGS ABOUT LINES
WHAT TO PROVE
HOW TO PROVE
FORMULA TO USE
Lines are PARALLEL
Lines are NOT PARALLEL
Lines are PERPENDICULAR
Lines BISECT EACH OTHER
Lines are CONGRUENT
1.) Prove that AB and CD bisect each other if
A(-1,-3) B(3,3)
C(-3,3) D(5,-3)
2.) Prove EF⊥GH if
E(2,-2) F(4,2)
G(1,1) H(5,-1)
3.) Prove that WX∥YZ if
W(1,1) X(5,4) Y(-2,-5) Z(2,-2)
4.) Prove that MN≅PO if
M(-8,1) N(-2,-1) O(3,2) P(1,-4)
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PROVING TRIANGLES/TRAPEZOIDS
1.) Given: R(1,-1), U(6,-1), N(1,-7).
Prove: RUN is a right triangle
2.)
What are we really trying to do? _______________________
How are we going to do that? __________________________
Come up with a plan. What are you
trying to do? How are you going to do it?
WRITE IT!!!
Given: R(-4,0), I(0,1), C(4,-1), K(-4,1)
Prove: RICK is a Trapezoid
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3.) Given: A(-6,9), B(10,-3) and C(-4,-5)
Prove: ABC is an isosceles right triangle
4.) Given: M(1,3), A(-1,1), R(-1,-2), K(4,3)
Prove: MARK is an Isosceles Trapezoid
13
PROVING PARALLELOGRAMS USING COORDINATE GEOMETRY
WAYS TO PROVE
How would we do this using coordinate
geometry?
1.) Prove that BOTH PAIRS of opposite sides are
2.) Prove that BOTH PAIRS of opposite sides are
3.) Prove that BOTH PAIRS of opposite angles are
4.) Prove that diagonals
5.) Prove that One Pair of opposite sides are both
and
Let’s pick a way and use it for all parallelograms!!!!!
1.) Prove that JOYS with vertices J(1,1) O(2,-6), Y( 3,3) S(2,10) is a parallelogram
2.) If three of the points of a parallelogram are (-1,2), (4,6), (6,0) find the other point. Justify your answer:
14
PROVING RECTANGLES USING COORDINATE GEOMETRY
WAYS TO PROVE
HOW ARE WE GOING TO DO THIS USING
COORDINATE GEOMETRY?
1.) Prove that it is a PARALLELOGRAM with
2.) Prove that it is a PARALLELOGRM with
3.) Prove that all angles are
Let’s pick a way and stick with it!
1.)Prove that SNOW with vertices S(-3,0), N(4,7),O(9,2), and W(2,-5)is a rectangle
2.) Find the coordinates of E if CHER is a Rectangle C(0,2) H(4,8) E(x,y) R(3,0). Justify your answer.
15
PROVING RHOMBUSES USING COORDINATE GEOMETRY
WAYS TO PROVE
HOW ARE WE GOING TO DO THIS USING
COORDINATE GEOMETRY?
1.) Prove that it is a PARALLELOGRAM with
2.) Prove that it is a PARALLELOGRAM with
3.) Prove that all sides are
Let’s pick a way and stick with it!!!!!
1.) Given NEWY with coordinates N(1,5) E(-1,2) W(1,-1) Y(3,2) prove NEWY is a rhombus
2.) If the coordinates of RHOM are R(-1,5), H(-4,1), O(-1,-3), M(x,y) . Find x &y. Justify your answer.
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PROVING A SQUARE USING COORDINATE GEOMTERY
WAYS TO PROVE
HOW ARE WE GOING TO DO THIS USING
COORDINATE GEOMETRY?
1.) Prove that it is a RECTANGLE with
2.) Prove that it is a RHOMBUS with
1.) Given YEAR with coordinates Y(1,-4) E(3,0) A (-1,2) R(-3,-2) prove YEAR is a square
2.) If SQUA is a square, find the coordinates of S(a,b) given Q(-4,-4), U(2,2), A(8,-4). Justify your answer.
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WHAT SHAPE?
1.)If quadrilateral ABCD has coordinates:
ABCD? Justify your answer.
, what type of quadrilateral is
2.) If quadrilateral ABCD has coordinates:
ABCD? Justify your answer.
, what type of quadrilateral is
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